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In multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.

There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar, while the cross product returns a pseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics, physics, and engineering. The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context.

To illustrate, consider three-dimensional Euclidean space, letting:

$\mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k}$
$\mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k}$

be two vectors where i, j, k (also denoted e1, e2, e3) are the standard basis vectors in this vector space (see also Cartesian coordinates). Then the dyadic product of a and b can be represented as a sum:

$\begin{array}{llll} \mathbf{ab} = & a_1 b_1 \mathbf{i i} & + a_1 b_2 \mathbf{i j} & + a_1 b_3 \mathbf{i k} \\ &+ a_2 b_1 \mathbf{j i} & + a_2 b_2 \mathbf{j j} & + a_2 b_3 \mathbf{j k}\\ &+ a_3 b_1 \mathbf{k i} & + a_3 b_2 \mathbf{k j} & + a_3 b_3 \mathbf{k k} \end{array}$

or by extension from row and column vectors, a 3×3 matrix (also the result of the outer product or tensor product of a and b):

$\mathbf{a b} \equiv \mathbf{a}\otimes\mathbf{b} \equiv \mathbf{a b}^\mathrm{T} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\begin{pmatrix} b_1 & b_2 & b_3 \end{pmatrix} = \begin{pmatrix} a_1b_1 & a_1b_2 & a_1b_3 \\ a_2b_1 & a_2b_2 & a_2b_3 \\ a_3b_1 & a_3b_2 & a_3b_3 \end{pmatrix}.$
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